Non-uniqueness of Leray Solutions to the Hypodissipative Navier–Stokes Equations in Two Dimensions

We exhibit non-unique Leray solutions of the forced Navier–Stokes equations with hypodissipation in two dimensions. Unlike the solutions constructed in Albritton et al. (Ann Math 196(1):415–455, 2022), the solutions we construct live at a supercritical scaling, in which the hypodissipation formally becomes negligible as t→0+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \rightarrow 0^+$$\end{document}. In this scaling, it is possible to perturb the Euler non-uniqueness scenario of Vishik (Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part I, 2018; Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part II, 2018) to the hypodissipative setting at the nonlinear level. Our perturbation argument is quasilinear in spirit and circumvents the spectral theoretic approach to incorporating the dissipation in Albritton et al. (2022).